Nope, as OS X finds the Printer and Audio equipment via "Bonjour/Rendezvous" it doesn't matter if you also have a 3rd party router. The problems occur when you, as you stated, try and WDS an Express with a 3rd party router. WDS is when two base stations work in tandem to increase the range of the wireless network. As you've said, you don't want to extend the range in that way, so you wont come across any WDS problems.
My worry however, is that if your 3rd party router is attached to your mac wirelessly then you wont be able to use the Airport express as a print or audio server, as they'll be on their own wireless network, generated by the Airport Express itself. If however, you are connected to your router through an ethernet cable AND your computer is airport/wireless compatible then you'll be able to be connected to both routers.
Another solution is to make the airport express your router by connecting its ethernet/wan port to your other router. Then you connect your computers to the internet VIA the Express and not your existing one. However, it would then appear you wasted money in the first place buying a wireless router.